On Non-Linear operators for Geometric Deep Learning
Authors: Grégoire Sergeant-Perthuis, Jakob Maier, Joan Bruna, Edouard Oyallon
NeurIPS 2022 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | This work studies operators mapping vector and scalar fields defined over a manifold M, and which commute with its group of diffeomorphisms Diff(M). We prove that in the case of scalar fields Lp ω(M, R), those operators correspond to point-wise non-linearities, recovering and extending known results on Rd. In the case of vector fields Lp ω(M, TM), we show that those operators are solely the scalar multiplication. Our first contribution is to demonstrate that the non-linear operators which act on vector fields (elements of Lp ω(M, TM)) and which commute with the group of diffeomorphisms, are actually just scalar multiplications. This implies that Diff(M) is too rich to obtain non-trivial operators. Our second contribution is to demonstrate that non-linear operators acting on signals in Lp ω(M, R) are point-wise non-linearities. The appendix contains complete formal arguments and technical lemmata which we omit here due to lack of space. |
| Researcher Affiliation | Academia | Grégoire Sergeant-Perthuis Univ. Artois, UR 2462, Laboratoire de Mathématiques de Lens (LML) F-62300 Lens, France & OURAGAN team, Inria Paris & IMJ-PRG Paris, France. gregoireserper@gmail.com Jakob Maier INRIA, DI/ENS, PSL Paris Joan Bruna Courant Institute of Mathematical Sciences New York University New York Edouard Oyallon MLIA Machine Learning and Information Access Sorbonne Université, CNRS, ISIR, F-75005 Paris, France |
| Pseudocode | No | The paper focuses on theoretical proofs and mathematical formalism and does not include any pseudocode or algorithm blocks. |
| Open Source Code | No | The paper is theoretical and does not mention any code release or provide links to source code for the described methodology. |
| Open Datasets | No | The paper is theoretical and does not involve empirical studies, datasets, or model training. Sections related to experiments are marked N/A in the authors' checklist. |
| Dataset Splits | No | The paper is theoretical and does not involve data splits (training, validation, test) or empirical experiments. |
| Hardware Specification | No | The paper is theoretical and does not describe any experimental hardware specifications. The authors' checklist explicitly marks this as N/A. |
| Software Dependencies | No | The paper is theoretical and does not list any software dependencies with version numbers for experimental reproducibility. |
| Experiment Setup | No | The paper is theoretical and does not describe an experimental setup, hyperparameters, or training settings. The authors' checklist explicitly marks this as N/A. |